Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Graph each line by hand. Give the \(x\) - and y-intercepts. $$2 x+5 y=10$$

Step-by-Step Solution

Answer

x-intercept: (5, 0), y-intercept: (0, 2)

Step 1: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation. The equation is \(2 \cdot 0 + 5y = 10\), which simplifies to \(5y = 10\). Solving for \(y\), we divide both sides by 5 to get \(y = 2\). Thus, the y-intercept is at the point \((0, 2)\).

Step 2: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation. The equation becomes \(2x + 5 \cdot 0 = 10\), simplifying to \(2x = 10\). Solving for \(x\), we divide both sides by 2 to get \(x = 5\). Thus, the x-intercept is at the point \((5, 0)\).

Step 3: Plot the intercepts

On a coordinate plane, plot the points \((0, 2)\) and \((5, 0)\) that represent the y-intercept and the x-intercept respectively. These points will help you draw the line.

Step 4: Draw the line

Using a ruler, draw a straight line through the points \((0, 2)\) and \((5, 0)\). Extend the line across the graph to show its path in both directions. This line represents the equation \(2x + 5y = 10\).

Key Concepts

X-intercept

In mathematics, the x-intercept is the point where a graph crosses the x-axis. This means that at this point, the value of y is zero. To find the x-intercept from a linear equation like the one given, you'll need to set y to zero and solve for x.

Let's look at the equation provided:

• Equation: \(2x + 5y = 10\)
• Set \( y = 0 \): \( 2x + 5 \cdot 0 = 10 \)
• This simplifies to \( 2x = 10 \)
• Solving for \( x \), divide both sides by 2, yielding \( x = 5 \)

Hence, the x-intercept is at the coordinate (5, 0). In simpler terms, when you graph this line, it crosses the x-axis at the point 5. Understanding x-intercepts helps in knowing where a line meets the horizontal axis.

Y-intercept

The y-intercept is another crucial point on a graph. It is where the line crosses the y-axis. At this interception, the value of x is zero. To find the y-intercept, you need to substitute x with zero in the linear equation and then solve for y.

Let's solve for the y-intercept in our example:

• Equation: \(2x + 5y = 10\)
• Set \( x = 0 \): \( 2 \cdot 0 + 5y = 10 \)
• This simplifies to \( 5y = 10 \)
• Solving for \( y \) involves dividing both sides by 5, resulting in \( y = 2 \)

So the y-intercept is at the point (0, 2). This means the line crosses the y-axis at 2. Knowing the y-intercept provides a preliminary location of the line's intersection with the vertical axis.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph equations. It is composed of two axes— the x-axis and the y-axis. These axes divide the plane into four quadrants, allowing us to pinpoint any location with precision.

Here’s what makes up the coordinate plane:

• The horizontal line is the x-axis.
• The vertical line is the y-axis.
• The point where these two lines intersect is called the origin, denoted as (0,0).

Graphing on a coordinate plane is essential for visualizing mathematical concepts. It allows us to plot points for intercepts and draw lines, like in the given example. This visual representation aids in understanding relationships between variables.

Straight Line

A straight line is the simplest form of linear equations representation on a graph. It shows the constant relationship between two variables. On a coordinate plane, a line can be easily drawn if we know at least two points it passes through, like the x-intercept and y-intercept.

Steps to graph a straight line:

• Identify intercepts or any two points on the line.
• Plot these points on the coordinate plane.
• Use a ruler to draw a line through these points, extending it across the plane.

For our equation \(2x + 5y = 10\), locating the points (0, 2) and (5, 0) allows us to accurately draw the straight line. This line graphically represents all the solutions of the equation. Understanding straight lines in graphs is fundamental in interpreting data and trends visually.